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Consistency of pion-pion phase shifts with analyticity
Volume 21 number 1
PHYSICS
LETTERS
15 April 1966
C O N S I S T E N C Y OF P I O N - P I O N P H A S E S H I F T S W I T H A N A L Y T I C I T Y L. LUKASZUK* and A. M A R T I N
CERN, Geneva
Received 4 March 1966
Rigorous inequalities on the pion-pion D wave scattering lengths are derived. They are shown to be important constraints in the analysis of experimental data.
R e c e n t l y , a p h a s e s h i f t a n a l y s i s of p i o n - p i o n s c a t t e r i n g has b e e n p r o p o s e d [1]. We w i s h to p o i n t out that the u s u a l a n a l y t i c i t y a s s u m p t i o n s , in p a r t i c u l a r the M a n d e l s t a m r e p r e s e n t a t i o n , but a l s o m u c h weaker a n a l y t i c i t y p r o p e r t i e s r e s u l t i n g f r o m a x i o m a t i c f i e l d t h e o r y [2] i n d i c a t e that the low e n e r g y b e h a v i o u r of the D w a v e p h a s e s h i f t s m a y h a v e to be c o r r e c t e d . The a n a l y t i c i t y p r o p e r t i e s r e s u l t i n g f r o m a x i o m a t i c f i e l d t h e o r y , w h i c h w i l l be u s e d h e r e a r e the following: the 7rolro ~ 7rolro a m p l i t u d e , w h i c h is the one w h e r e the c o n t r a d i c t i o n m i g h t o c c u r , has the f o l l o w i n g p r o p e r t i e s [2]: 1) Fyoro(S , t, u) with s = s q u a r e of c . m . e n e r g y f o r s > 4 p 2, t = - 2 k 2 ( 1 - c o s 0), k m o m e n t u m , 0 s c a t t e r i n g a n g l e , and s + t+ u = 4 # 2, s a t i s f i e s f i x e d t d i s p e r s i o n r e l a t i o n s , not only f o r t ~< 0 but a l s o f o r ]t I < 4 ~ 2 ; 2) the s c a t t e r i n g a m p l i t u d e f o r s r e a l > 4p 2 can be p r o l o n g e d f r o m t = 0 to t = 4~ 2, i n s i d e a c o m p l e x n e i g h b o u r h o o d of the r e a l t a x i s 3) due to c r o s s i n g s y m m e t r y t h i s h o l d s f o r any p e r m u t a t i o n of s, t, u. Now f r o m t h e s e a s s u m p t i o n s , J i n and one of the a u t h o r s [3] d e d u c e d t h a t f i x e d s d i s p e r s i o n r e l a t i o n s hold with two s u b t r a c t i o n s f o r 0 --< s < 4 ~ 2, i . e . , b e l o w the s t h r e s h o l d . T h i s m e a n s that the F r o i s s a r t G r i b o v r e p r e s e n t a t i o n [4] f o r p a r t i a l w a v e s h o l d s f o r l >i 2, 0 ~< s < 4p2:
f~2O%(s)_
4
~ F 2t' ] f At(s,t')422L~-_-_~ljdt'
7r(4- s) 4
(1)
where u- t crossing symmetry has been used and A t is the absorptive part in the t channel. Now, notice that At(s , t') is positive for 0 --< s --< 4~ 2 because its Legendre polynomial expansion in the t channel converges, from assumptions 1) and 2) and has positive coefficients, due to unitarity. In the neighbourhood of s = 4~ 2, Q2 can be replaced by its asymptotic expression:
fn2oTro(s)
( s - 4 ) 2 tF~ [ ° ° A t ( s , t ' ) d t ' 2~/~- r(~) ~ (2t,)3
•
(2)
Now two things can happen: either the integral has finite positive limit as s -~ 4p 2 or it goes to +oo. The second possibility can be excluded if the S waves have normal threshold behaviour, by using fixed t dispersion relations and elastic unitarity of the T = 0 and T = 2 components. The argument is rather lengthy and will be presented elsewhere. The conclusion therefore is that if the S waves have normal threshold behaviour
f 2°%(s)
lim - s ~ 4 ( s - 4) 2 e x i s t s mad is
positive
[5].
• On leave of absence from the institute for Nuclear Research, Warsaw, Poland. 96
Volume 21, number 1
PHYSICS LETTERS
15 April 1966
½fT=O(s) + ~f T=2(s)
Therefore lim s--4
~ 0 .
(3)
(s- 4)2
This condition is not obviously violated by the analysis of ref. 1 in the sense that the D wave phase shifts are too small to be estimated in the region 0.28 GeV < ~/s < 0.58 C,eV. However, from 0.58 GeV I o + ~62 22 is negative and becomes later on positive for ~/s > 1 C,eV. Thereto 1 GeV, the combination ~62 i o 22 fore if one sticks to the analysis, the only way to reconcile it with the theory is to assume that ~62 + ~ 2 vanishes twice in the interval 0.28 < ~/s < 1.2 C,eV. This, we think, is difficult to believe because one I RedO .:2 to stay positive for some time above threshold. Indeed the representation (1), expects ~ -'2 + _32Re J2 if there is anything true in the application of Regge's ideas to elementary particle scattering (remember that here we are in the low energy region where the potential description can make sense), is expected to hold as long as we d o n o t m e e t a J = 2 , T = 0 o r T = 2 resonance. Then in (1) the nearest part of the integrand dominates, and we know from unitarity that At(s ,t') is positive for t' close to 4p 2 (at s = 16~ 2, At(s,t') is certainly positive up to t' = 1692). So we believe that the "signs of the D waves should be slightly improved". To conclude, let us indicate some more stringent conditions on the D wave scattering lengths once t h e S a n d P w a v e s a r e k n o w n u p to s o m e e n e r g y s o. T h e s e c o m e f r o m t h e p o s i t i v i t y p r o p e r t i e s of t h e a b s o r p t i v e p a r t s a n d f r o m t h e p o s i t i v i t y of c e r t a i n c r o s s i n g m a t r i x c o e f f i c i e n t s : i O 2 2 8(~f2(s) +~/2(s)) -3a2~ o + ~a 222 = l i m s-~4
1)
2)
8:F°(s)
>
(s- 4) 2
8r(3)
r
J s° 2~/~ F({) 4
2 ~ ~ f o O(s), + ~[mf2(s') (~s,~3
ds'
8r(3)
a~ = lim > - s--.4 (s- 4)2 2 J ~ F(½) 4
If o n e i n s e r t s i n t o t h e s e s u m r u l e s t h e n u m e r i c a l t h e i n t e g r a l a t s o = 1.40 GeV, o n e g e t s
as' (2s') 3
"
values proposed by Wolf for S and P waves,
]lao 2 + ~2a2 2 > 0.0013 - -1 ~5'
a~ > 0.0021
c u t t i n g off
~5"
The most important contribution comes from the threshold bump in the T = 0 S wave. One of u s ( L . ~ . ) t h a n k s P r o f e s s o r s Study Division.
L. V a n Hove a n d J. P r e n t k i f o r h o s p i t a l i t y a t t h e C E R N T h e o r e t i c a l
References 1. 2. 3. 4.
G.Wolf, p r e p r i n t DESY, Hamburg. A. Martin, CERN p r e p r i n t TH. 637 (1965), to appear in Nuovo Cimento. A. M a r t i n and Y. S. Jin, Phys. Rev. 135 {1964) B1375. M . F r o i s s a r t , La Jolla Conference, 1961 (unpublished); V . N . G r i b o v , J . E x p t l . T h e o r e t . P h y s . (USSR) 41 (1962) 1961; A . M a r t i n , P h y s i c s L e t t e r s 1 {1962) 72; E . J . S q u i r e s , Nuovo Cimento 25 (1962) 242. 5. A s i m i l a r r e s u l t has been obtained independently, s t a r t i n g f r o m Mandelstam r e p r e s e n t a t i o n by G. Wanders, P h y s i c s L e t t e r s 19 (1965) 331.
97
PHYSICS
LETTERS
15 April 1966
C O N S I S T E N C Y OF P I O N - P I O N P H A S E S H I F T S W I T H A N A L Y T I C I T Y L. LUKASZUK* and A. M A R T I N
CERN, Geneva
Received 4 March 1966
Rigorous inequalities on the pion-pion D wave scattering lengths are derived. They are shown to be important constraints in the analysis of experimental data.
R e c e n t l y , a p h a s e s h i f t a n a l y s i s of p i o n - p i o n s c a t t e r i n g has b e e n p r o p o s e d [1]. We w i s h to p o i n t out that the u s u a l a n a l y t i c i t y a s s u m p t i o n s , in p a r t i c u l a r the M a n d e l s t a m r e p r e s e n t a t i o n , but a l s o m u c h weaker a n a l y t i c i t y p r o p e r t i e s r e s u l t i n g f r o m a x i o m a t i c f i e l d t h e o r y [2] i n d i c a t e that the low e n e r g y b e h a v i o u r of the D w a v e p h a s e s h i f t s m a y h a v e to be c o r r e c t e d . The a n a l y t i c i t y p r o p e r t i e s r e s u l t i n g f r o m a x i o m a t i c f i e l d t h e o r y , w h i c h w i l l be u s e d h e r e a r e the following: the 7rolro ~ 7rolro a m p l i t u d e , w h i c h is the one w h e r e the c o n t r a d i c t i o n m i g h t o c c u r , has the f o l l o w i n g p r o p e r t i e s [2]: 1) Fyoro(S , t, u) with s = s q u a r e of c . m . e n e r g y f o r s > 4 p 2, t = - 2 k 2 ( 1 - c o s 0), k m o m e n t u m , 0 s c a t t e r i n g a n g l e , and s + t+ u = 4 # 2, s a t i s f i e s f i x e d t d i s p e r s i o n r e l a t i o n s , not only f o r t ~< 0 but a l s o f o r ]t I < 4 ~ 2 ; 2) the s c a t t e r i n g a m p l i t u d e f o r s r e a l > 4p 2 can be p r o l o n g e d f r o m t = 0 to t = 4~ 2, i n s i d e a c o m p l e x n e i g h b o u r h o o d of the r e a l t a x i s 3) due to c r o s s i n g s y m m e t r y t h i s h o l d s f o r any p e r m u t a t i o n of s, t, u. Now f r o m t h e s e a s s u m p t i o n s , J i n and one of the a u t h o r s [3] d e d u c e d t h a t f i x e d s d i s p e r s i o n r e l a t i o n s hold with two s u b t r a c t i o n s f o r 0 --< s < 4 ~ 2, i . e . , b e l o w the s t h r e s h o l d . T h i s m e a n s that the F r o i s s a r t G r i b o v r e p r e s e n t a t i o n [4] f o r p a r t i a l w a v e s h o l d s f o r l >i 2, 0 ~< s < 4p2:
f~2O%(s)_
4
~ F 2t' ] f At(s,t')422L~-_-_~ljdt'
7r(4- s) 4
(1)
where u- t crossing symmetry has been used and A t is the absorptive part in the t channel. Now, notice that At(s , t') is positive for 0 --< s --< 4~ 2 because its Legendre polynomial expansion in the t channel converges, from assumptions 1) and 2) and has positive coefficients, due to unitarity. In the neighbourhood of s = 4~ 2, Q2 can be replaced by its asymptotic expression:
fn2oTro(s)
( s - 4 ) 2 tF~ [ ° ° A t ( s , t ' ) d t ' 2~/~- r(~) ~ (2t,)3
•
(2)
Now two things can happen: either the integral has finite positive limit as s -~ 4p 2 or it goes to +oo. The second possibility can be excluded if the S waves have normal threshold behaviour, by using fixed t dispersion relations and elastic unitarity of the T = 0 and T = 2 components. The argument is rather lengthy and will be presented elsewhere. The conclusion therefore is that if the S waves have normal threshold behaviour
f 2°%(s)
lim - s ~ 4 ( s - 4) 2 e x i s t s mad is
positive
[5].
• On leave of absence from the institute for Nuclear Research, Warsaw, Poland. 96
Volume 21, number 1
PHYSICS LETTERS
15 April 1966
½fT=O(s) + ~f T=2(s)
Therefore lim s--4
~ 0 .
(3)
(s- 4)2
This condition is not obviously violated by the analysis of ref. 1 in the sense that the D wave phase shifts are too small to be estimated in the region 0.28 GeV < ~/s < 0.58 C,eV. However, from 0.58 GeV I o + ~62 22 is negative and becomes later on positive for ~/s > 1 C,eV. Thereto 1 GeV, the combination ~62 i o 22 fore if one sticks to the analysis, the only way to reconcile it with the theory is to assume that ~62 + ~ 2 vanishes twice in the interval 0.28 < ~/s < 1.2 C,eV. This, we think, is difficult to believe because one I RedO .:2 to stay positive for some time above threshold. Indeed the representation (1), expects ~ -'2 + _32Re J2 if there is anything true in the application of Regge's ideas to elementary particle scattering (remember that here we are in the low energy region where the potential description can make sense), is expected to hold as long as we d o n o t m e e t a J = 2 , T = 0 o r T = 2 resonance. Then in (1) the nearest part of the integrand dominates, and we know from unitarity that At(s ,t') is positive for t' close to 4p 2 (at s = 16~ 2, At(s,t') is certainly positive up to t' = 1692). So we believe that the "signs of the D waves should be slightly improved". To conclude, let us indicate some more stringent conditions on the D wave scattering lengths once t h e S a n d P w a v e s a r e k n o w n u p to s o m e e n e r g y s o. T h e s e c o m e f r o m t h e p o s i t i v i t y p r o p e r t i e s of t h e a b s o r p t i v e p a r t s a n d f r o m t h e p o s i t i v i t y of c e r t a i n c r o s s i n g m a t r i x c o e f f i c i e n t s : i O 2 2 8(~f2(s) +~/2(s)) -3a2~ o + ~a 222 = l i m s-~4
1)
2)
8:F°(s)
>
(s- 4) 2
8r(3)
r
J s° 2~/~ F({) 4
2 ~ ~ f o O(s), + ~[mf2(s') (~s,~3
ds'
8r(3)
a~ = lim > - s--.4 (s- 4)2 2 J ~ F(½) 4
If o n e i n s e r t s i n t o t h e s e s u m r u l e s t h e n u m e r i c a l t h e i n t e g r a l a t s o = 1.40 GeV, o n e g e t s
as' (2s') 3
"
values proposed by Wolf for S and P waves,
]lao 2 + ~2a2 2 > 0.0013 - -1 ~5'
a~ > 0.0021
c u t t i n g off
~5"
The most important contribution comes from the threshold bump in the T = 0 S wave. One of u s ( L . ~ . ) t h a n k s P r o f e s s o r s Study Division.
L. V a n Hove a n d J. P r e n t k i f o r h o s p i t a l i t y a t t h e C E R N T h e o r e t i c a l
References 1. 2. 3. 4.
G.Wolf, p r e p r i n t DESY, Hamburg. A. Martin, CERN p r e p r i n t TH. 637 (1965), to appear in Nuovo Cimento. A. M a r t i n and Y. S. Jin, Phys. Rev. 135 {1964) B1375. M . F r o i s s a r t , La Jolla Conference, 1961 (unpublished); V . N . G r i b o v , J . E x p t l . T h e o r e t . P h y s . (USSR) 41 (1962) 1961; A . M a r t i n , P h y s i c s L e t t e r s 1 {1962) 72; E . J . S q u i r e s , Nuovo Cimento 25 (1962) 242. 5. A s i m i l a r r e s u l t has been obtained independently, s t a r t i n g f r o m Mandelstam r e p r e s e n t a t i o n by G. Wanders, P h y s i c s L e t t e r s 19 (1965) 331.
97